Find intersection of linear and logarithmic linesI have equations for two lines, one of which is linear and the other is logarithmic, ie:y=m1x+c1y=m2⋅ln⁡(x)+c2..and I need to find out where (if at all) these lines intersect. I realise that I need to solve:m1⋅x+c1=m2⋅ln⁡(x)+c2..for x, but apart from shuffling the constants around I'm not sure how to do thisIs there a general solution to this problem?Thanks

Question

Find intersection of linear and logarithmic linesI have equations for two lines, one of which is linear and the other is logarithmic, ie:y=m1x+c1y=m2⋅ln⁡(x)+c2..and I need to find out where (if at all) these lines intersect. I realise that I need to solve:m1⋅x+c1=m2⋅ln⁡(x)+c2..for x, but apart from shuffling the constants around I&#039;m not sure how to do thisIs there a general solution to this problem?Thanks

muthe2ulj · Accepted Answer

The general solution involves the Lambert W function, whose defining equation is z=W(z)eW(z) for complex numbers z. If either m1 or m2 is zero in the given problem, then the solution is elementary, so suppose m1,m2≠0. Thenm1x+c1=m2ln⁡x+c2can be rewritten as−m1m2ec1−c2m2=−m1xm2e−m1xm2,which has the solution−m1xm2=W(−m1m2ec1−c2m2),orx=−m2m1W(−m1m2ec1−c2m2).For instance, if m1=1,m2=−1,  and  c1=c2=0 this gives x=W(1)=Ω=0.56714329… (the Omega constant), which is correct, since Ω=−ln⁡Ω

Find intersection of linear and logarithmic lines I have

Answered question

Answer & Explanation

New Questions in Algebra II